Download Analysis of the detective quantum efficiency of

Proc. of SPIE Vol. 1896, Medical Imaging 1993: Physics of Medical Imaging, ed. R Shaw (Sep 1993) Copyright SPIE
Analysis of the detective quantum efficiency of coupling a CCD
to a scintillating phosphor for x-ray microtomographic
imaging
M.S. Westmore’and I.A. Cunningham’,’
‘Imaging ResearchLaboratories,John P. Robarts ResearchInstitute,
and Department of Medical Biophysics, University of Western Ontario,
100 Perth Drive, London, Ontario, CANADA, N6A SK8
“Departmentof Radiology, Victoria Hospital
London, Ontario, CANADA
ABSTRACT
We are developing an x-ray microtomographic imaging system @CT) for imaging small objects at very high (-25 pm)
spatial resolution. The detector for this system consists of a CCD array coupled to a phosphor screen through a fiber-optic
faceplate.For the purposesof signal and noise analysis, this system is modeled as a multi-stage cascadedimaging system
consistingof: a) conversionof x-ray quantato optical quantain the phosphor;b) collection and transfer of optical quanta from
the phosphorto the CCD; and c) detection of optical quanta by the CCD. We use the model of Rabbani et al.’ for cascaded
systemsto theoretically calculate the detective quantum efficiency (DQE) as a function of spatial frequency.
Proper engineeringdecisionsin the design and optimization of cascadedsystemsrequires a knowledge of quantum fluence
levels at each stage.Conventional nomogramshave beenused in the past to display this information graphically, and to ensure
that the “quantum sink” occurs at the x-ray detection stage.However, this approachneglectsthe spatial spreadingof secondary
quantain the cascadedsystem, and hence is limited as being a zero-frequencyanalysis.
We have developedthe theoretical basis of a spatial-frequencydependentnomogram in terms of the system DQE. This
approachis used to identify any sourcesof image degradation,and to m‘akeoptimal design decisionsof systemparameterssuch
as optical gains or numerical apertures.Using this approach,we show that the spreadingof optical photons in the phosphor
screen is the most significant factor degrading the MTF. We show that a 40 pm phosphor has a 10% MTF value at
approximately 25 mm-‘, which is well matched to the 21 mm-’ sampling cutoff frequency of the CCD. The conventional
nomogramanalysis indicates this combination is quantum limited at the x-ray detection stage. However, the spatial-frequency
dependentnomogram shows this is a misleading result, and indicates that for frequenciesabove approximately 15 mm”, the
system is “quantum limited” by the optical photon fluence. An additional gain factor of 10X in the optical system is required
to remove this limitation.
1. INTRODUCTION
We are developing a special-purposex-ray microtomographic(pCT) imaging system for the study of researchanimals and
other specimens.When designingor optimizing imaging systemssuch as this, it is necessarylo know the quantumfluence levels
at each stage of the system to make proper choices of system gain factors. or other parameterssuch as optical numerical
apertures.One approachused in this type of analysis, is to representthe systemas a series of cascadedstages.The number of
quantain one pixel or minimum resolving element is determinedfor each stageas the product of gains and efficiencies of all
precedingstages.This information can be displayedin a system nomogram,showing the number of quantaon the vertical ‘axis
for each stage.2”The stage with the fewest quanta (N,,,,) is called a “quantum sink”,” and ultimately limits the image signal-tonoise ratio (SNR) to the squareroot of N,,. System designerscan use the nomogramto ensure that the quantum sink occurs
at the x-ray detection stage.In this model, additive system noises are not included.
Inherent in this approach,however, is the assumptionthat all secondaryquantacorrespondto the sameposition in the final
image as the primary interacting quanta.This neglectsscatterprocesseswhich causea spreadingof secondnryquanta.Thus, the
conventional nomogramis a zero-spatialfrequencyanalysis, and is not sensitive to the spatial-frequencycharacteristicsof the
system.
82 / SPlE Vol. 189G Physics of Medical imaging
(1993)
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In this paper, we extend the concept of the conventional nomogram to include spatial-frequencydependence,and apply the
result to an analysis of the microtomographic detector. A quantitative and physical interpretation of the spatial-frequency
dependentnomogram is obtained by expressing it in terms of the spatial-frequencydetective quantum efficiency (DQE). The
DQE is determined by representing the detector as a number of cascaded stages consisting of amplifying or scattering
mechanisms.
2. SYSTEM
DESCRIPTION
The pCT system is shown schematically in Fig. 1. It utilizes a conventional diagnostic x-ray tube and a specially designed
high-resolution x-ny detector. A schematicof the detectoris shown in Fig. 2. It consistsof a CCD (Tektronix TK1024) optically
coupled to a scintillating phosphor screenthrough a fiber-optic faceplate. The CCD is a 1024X1024 element array of 24 pm by
24 pm pixels. The detector assembly is placed approximately 75 cm from the x-ray tube. A rotating specimen stage is placed
directly in front of the detector.
In order to analyze the propagation of signal and noise through the system, we model the detector as a multi-stage cascaded
imaging system, consisting of: a) conversion of x-ray quanta to optical quanta in the phosphor; b) collection and transfer of the
optical quanta from the phosphor to the CCD; and c) detection of the optical quanta by the CCD. We further divide this process
into the seven stagesdescribed later in Table I. In the following section, expressionsthat describe the performance of each of
these seven stagesare derived. These ‘are then used in an evaluation of signal and noise propagation.
3. THEORY
3.1. Background
We characterize our system in terms of the spatial-frequencydependentDQE, based on the model of Rabbani et. al.’ for
cascadedsystems.They use multivariate moment-generatingfunctions to derive relationships between the input and output noise
power spectra at each stage. We let 0,,, and S,,(f) describe the mean quantum fluence and the corresponding noise power
spectrumrespectively that are input to the i” stage of the system, and 0; and S,(f) describe the output. Three types of processes
must be considered when describing the propagation of signal and noise through a cascaded system. These are: a) bin‘ary
selection; b) scattering; and c) amplification.
a) Binary Selection Processes
In a binary selection process,there is a probability p (p 5 1) of a given input quantum appearing at the output of the stage.
For an individual quantum, the outcome can be 1 or 0 only. The signal and noise power spectrum are given by
(1)
ai = I)@),-,
S;(f) = ~‘2S;_,(f)+p(l-~‘)~;-,
In the limit of small p, S,(f) = p<p., = oi, which indicates that the output is Poisson distributed and uniform in frequencies
rotate specimen
stage
CCD
x-rayfl:
=
2-D detector
data acquisition
computer
,0,
Figure 1. Schematic of the pCT system.
Fiber Optic
Faceplate
Figure 2. Schematic of the pCT delector.
SPIE Vol. 7 896 Physics of Medical
Imaging
(I 993) / 83
regardlessof the shape of the input distribution, while as p -+ 1, ai -+ CD,,and S,(f) -+ S,.,(f), as expected.
b) Scattering Processes
A scatteringprocess is characterizedby the modulation transfer function of the stage,T(f), which is equal to the magnitude
of the Fourier transform of the scatter line spread function normalized to unity at zero frequency. The signal and noise
propagation are given by
(3)
Qi = oi-,
‘i(f)
= (‘i_l(f)-~i_,
(4)
)T2(f)+~j-,
If T(f) = 1 for all frequencies,corresponding to ‘an impulse LSF and hence no scattering, then S,(f) = S,.,(f), as expected.
c) Amplification Processes
An amplification processis one in which there are more output quanta than input quanta. The mean gain is given by g and
the variance in the gain by 0,“. The variance can be used to include the effect of terms such as Swank noise.“’ The propagation
of signal and noise through such a process is governed by the following equations.
(5)
pi = ,$~,_I
(6)
S;(f) = g’s;-, (f‘)+p;-,
If the gain is deterministic (crg2= 0), then S,(f) = g’S,,(f). as expected.
Using these equations,it is possible to characterizean entire system,requirin,(7only the gains, MTFs. and o,s for each stage.
In the following sections, we determine these parametersfor each stage.
3.2. MTF Calculation
The optical scatterLSF is determinedusing Swank’s model%for a transparentphosphor with exponential absorptionof x-rays
and a black backing. We modify this model to include the effect of an index-matched protective overcoat on the screen. In
addition, light scattering tow‘ardsthe tails of the LSF exit the overcoat at an increasedangle from the normal (0,, in Fig. 3). This
reducesthe coupling efficiency into the fiber due to the fiber’s non-matchedindex of refraction. A simiharprocessoccurs at the
fiber-silicon interface on the CCD. As a result, it is necessaryto combine these effects into a single phosphor-fiber-silicon LSF.
As representedschematically in Fig, 4, the phosphor screenhas thickness D and is contained between z = D, and z = D, + D,
where D, is the thickness of the overcoat. The fiber-optic bundle extendsin the
negative z-direction below z = 0. The screen is assumedto be infinite in the x
and y-directions.
Optical
Burst
z
.
v
BP n
Phosphol
Screen
Overcoat
The LSF is determined as the detector responseto a thin line-beam of x-rays
incident from z = +OOin the yz plane. The detector responseis determined by
calculating the number of quanta that are emitted from element dydz that pass
through clement dA and are then transmitted first from the phosphor to the fiber
and then from the fiber to the silicon. The result is integrated over z from D, to
D, + D and over y from -M to 00.The LSF is therefore
I)~,+fJ
,.>
L,,(x ) = ;
* 3ev(-pl(o,,+Z)
JJ -,X1
47cl.j
I)”
Fiber Optic
Faceplate
where
CC0
:
: ‘,
Silicon
:
: ‘,
L-ys
I I
II’
’
e8
A
Figure 3. Coupling of optical photons
between phosphor and CCD.
84 I SPIE Vol. I896
Physics of Medical
imaging
(I 993)
>-z I)~~;,(~,,,~,)E,~(e,es)~ly~/2,
(7)
C = constant required to normalize L,,(x) to unit area,
E,,, = the transmittancefrom the phosphor to the fiber,
E, = the transmittancefrom the fiber to the silicon,
9,, = the angle of the light ray to the normal in the phosphor,
0, = the angle of the light ray to the normal in the fiber,
8, = the angle of the light ray to the normal in the silicon.
The light is unpolarized and therefore we must separatethe two perpendicularly
polarized components, evaluate the above expressionfor each, and average the
results. The perpendicular and parallel transmittancesare given by9
E, =
E, =
4n,ri, cos0,cos0,
(noCOSe”+n,cose,)’
4n,n,cos8,costl,
(n, cos0,+n,,cos~, )’
for a wave passing from material 0 to material 1, with angles 8,, and 8, to the
normal. The angles 8, and 8, are dependentby Snell’s law upon the indices of
retraction of the phosphor (n,), fiber core (nJ, and silicon (n3. The index of
refraction of the fiber cladding (n,) also has an effect on the LSF since the
numerical aperture of the fiber defines a maximum angle of incidence 6,,““’
beyond which no light is transmitted, given by”
flpmax= sin-1 ~~
r “p
Geometry for calculating the
scatter LSF for coupling between the
phosphor, fiber optics, and silicon in the
CCD.
1
The resulting combined MTF, T,,(f), is given by the magnitude of the Fourier
transform of L,(x). It should be noted that the effect of fl,,““’as well as the fact
that the transmittancesdecreaseas a function of ‘angle,actually reducesthe LSF width, and therefore improves the MTF at the
expenseof light transmission efficiency.
The MTF due to the aperture function of the fiber, T,(f), is approximated using a one-dimensional simplification of the
aperture by assuming it to be a rectangle of width a,, where a( is the diameter of the fiber core. Therefore,
sin (rc(ll.f‘)
(10)
T,(f) = sinr:(m/f) =
mpf
(
This simplification is justified on the grounds that a, = 4.7 pm and hence T,(f) has a negligible effect on the overall MTF.
The edge responsefunction (ERF) of the CCD was measuredusing a slantededge technique” where an opaque metal blade
was illuminated by a uniform field of green light. The CCD MTF, T,(f), was calculated from this ERF data. The effect of the
MTF of the lens used to focus the light was removed by dividing the measuredMTF by the lens MTF which was measured
separately.i2
3.3. Other System Parameters
The x-ray detection quantum efficiency, rl, of the phosphor screen is given approximately by
7 = I-exp(-VT),
(11)
where p is the linear attenuation coefficient of the screen.
The mean optical gain of the phosphor is denoted by m and the variance in the gain by o,,,~.In addition, the v‘ariancecan
be expressedas a Poisson excessterm, given by
Note that if the gain is Poisson distributed, on,’= m, and E = 0.
The geometry for calculating t, the transmissionefficiency from the phosphor to the silicon, is shown in Fig. 5. We assume
that when an x-ray is detected in the screen, optical photons are emitted isotropically. The solid angle d!2 is given by
SPlE Vol. 1836 Physics of Medical
lmaging
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I
Stage #
Description
Type of process Gain / Efficiency
1
Absorption of x-rays in phosphor
bin‘ary selection
l-l
---
2
Emission of photons in phosphor
amplification
---
3
Phosphor-fiber-siliconMTF
scattering
g
___
T,,(f)
4
Transmissionfrom phosphorto silicon
binary selection
t
-__
5
MTF due to fiber aperture function
scattering
---
T,(f)
6
MTF due to CCD aperturefunction
scattering
---
T,(f)
7
Detection in the CCD
binary selection
a
---
M-W-l
Table I. The stages in the pCT system.
(13)
dC2 = 27csin0&.
phosphor
I\
The transmission efficiency is determined by integrating the
product of the transmittancefrom the phosphorto the fiber and
the transmittancefrom the fiber to the silicon over all solid
angle elements d&X,dividing by 4~~ and multiplying by the
packing fraction, PF, of the fiber-optic faceplate.This gives
t = g
e~d2EJOp,e,,E,
(14)
s (0,,f$).
0
-
*.*.
‘.*.
‘.
-A
Figure 5. Geometryfor calculating efficiency of transmission
from phosphorto CCD.
This integral is also evaluated for the two polarization
components.
3.4. Noise Power Spectrum of Detector
The quantum fluence and noise power spectrumat eachstageare calculatedwith Eqs. (l)-(6), and summarizedin Table II.
3.5. Detective Quantum
Efficiency of Detector
The spatial-frequencydependentDQE of an imaging system is defined as
DQE(f)
=
.SNR,>cf)
(15)
SfVf?;‘cf)’
where SNR,(f) and SNRi(f) are the frequency-dependent
output and input signal-to-noiseratios respectively.If MTF(f) is the
total system modulation transfer function and S,(f) and S,(f) ‘arethe output and input noise power spectrarespectively, then
oQEcf>
=
@<,‘MTF ‘(f)/Sc2(f)
oi ‘/S,(f)
(16)
.
If the total system gain is g and the input is Poissondistributed (S,(f) = OJ, then
8B / SPlE Vo!. 1896 Physics of Medical
Im;,,ging (1993)
STAGE
-------
PROCESS
----------
NOISE -POWER SPECTRUM
----__---_________________
QUANTUM FLUENCE
----_
_-----_-----
0
qf)
00 = @,
0,
1
binary selection
2
amplification
3
scatter
4
binary selection
5
scnttel
0’s
6
scatter
0,
0,
11q
=
mo,,
=
ml-p,
=
a*
=
ml-pi
=
U-D,,
=
tn11yD,
=
04
=
tnzlyDi
=
*s
=
mlp,,
a3
o4
S,(f) = q2som + q(l-W”
= l-g
= I@,,
=
= 0,
S,(f) = m*s,(f> + o;o,
= n22lJ(l+$zi
S,(f) = (S,(f) -@JT;(f)
S‘Jf) = t”S,(f)
= twq(1
7
binary selection
a@,,
=
aln?1p,
+ rnrz~-@~
+ t( 1 - tp,
+L)qT;(f)
m
= (qf)-qT;(f)
+ /mIpi
+ oD,
= l%?‘~( 1 +$a+7;(f)I;(f.)
S,,(f) = (S,(f) -qT<?v‘)
=
+ 0,
= nz%j(l +;)@;T,;(f)
.$(f)
a+
+ f?Z?-jQi
+ tnzlp;
+ OS
= t*m’17( 1 +;)Q,T;(f)T;(f)T;y.)
+ tnqOi
S,(f) - rx2S,(f’) + lx( 1 -a)0,
Table II. Quantum fluence and noise-power spectrum of ench strrge of thr sr:reerl-fi’be).-CCD system.
g”Q;‘MTF
DQEV)
=
yf)/sJ)
(17
0; 2/q
For the pCT detector, the total gain and MTF are given by
(q = atm11,
Gus
(l@
= T,,(f)T,(f)T<.(f’).
(19)
Therefore, the output noise power spectrumcan be expressedby (stngc 7 of Table 11)
S,,V)= -iix2I+m& cy477F’V’)+gay
t 1
w
Combining Eqs. (17)-(20), the DQE for the pCT detectorsystem can be written,
SPlE Vol. 1036 Physics of Medical Imaging
(1993) / 87
(21)
where
P(f)
(22)
= atnzllT,*V)T,“(f)T~‘(f)
is the product of the total system gain and MTF squared.
In Eq. (21), E is the Poissonexcessof the optical gain distribution. which is related to the Swank I facto8v6.’
of a phosphor
screen,and can be written
, -_ -. 4’
(23)
4?2
&, M,, and M, are the first three moments of the gain distribution, given byI M,, = 1, M, = m, and M2 = m2 + o,,,* =
m* + m(1 + E). In the current study, m = 1319,14and henceE / m can be written approximately as
- E =--.l I
(24)
m
3.6. Spatial-Frequency
I
Dependent Nomogram
Using the sameapproachused to develop Eq. (21), it can be shown that the DQE at the 11"' stageof the system is given by
I1 = 1
DQE,,(f) = 11,
7
=
11 ’
m p,,
(25)
n > 1
I+I+
where P,(f)
..- is the product of the gains and T*(f) of eachstage up to and including the II"' stage.The physical internretationof
P,(f) can be obtained by comparing Eq. (25) with the corre~ponhing expressionfor the zero-frequencyDQE:
DQEn = 11,
=
I1 = 1
11
I+
11.
l+
wl
(26)
/I > 1
T
where P, is the product of the gains of each stage up to and
including the nth stage. Eq. (26) is particulnrly useful becauseit
defines the relationship between the conventional zero-frequency
nomogramP, and the DQE, allowing for a quantitativeinterpretation
of the nomogram. It can be seen that P, quantifies the DQE
degradation due to the finite number of quanta at each stage
following detection of the primary x-ray quanta. For instance, a
value of P, >> IJ for all n will have a negligible effect on the DQE,
while P, = 77at some stage will reduce the DQE by approximately
half (assumingE / m << 1) at that stage,forming a “quantum sink”.
P,(f) has the appearance,therefore, of a spatial-frequency
dependentnomogram.It reducesIhc spalinl-frecluency
DQE in Ihe
same manner as P, reduces the zero-frequencyDQE, and cm be
‘[
Stage#, n
P”(f)
1
T
2
Yg
3
UT,:-(9
4
W-&,“(f)
5
qgtT,,‘(t)T;(t)
6
WT,,‘O-K%V,*U~
7
~gtcxT,,‘(f)T;(f)T,‘(f)
interpretedas the effective number of secondaryquanlapropagating Table III. P,,(f) at each stage.
88 / SPlE Vol. 1896 Physics of Medical imaging (1993)
the image signal through the system at each frequency. Expressionsfor P,(f) at each stage are shown in Table III.
4. RESULTS
4.1. MTFs
Fig. 6 shows the constituent MTFs and the total system MTF assuminga 40 pm Gd,O,S screen,a 10 pm overcoat, and 20
keV incident x-rays. This figure clearly demonstratesthe dominant effect of the spreadingof the optical photons in the phosphor
screen.Fig. 7 shows the effect of the overcoat thickness on the MTF of the phosphor-fiber-silicon combination. Clearly, this
effect is also quite important.
4.2. Other System Parameters
Gd,O,S has a density of 7.44 g/cm3,14and using the attenuation coefficients of the three constituent atoms,” the linear
attenuation coefficient of the screen is calculated’6 to be 274 cm-’ for 20 keV x-rays. Using this value leads to the quantum
efficiency values in Table IV.
Screen thickness (pm)
Gd,O,S has a peak emission wavelength of 545 nm and ‘an
intrinsic energy conversion efficiency of 1S%.14This leads to a
conversion gain factor of m = 1319 for a 20 keV incident x-ray.
Quantum efficiency 11
‘*i
The index of refraction of the screen is approximately
np = 1.5, while the index for silicon at 545 nm is n, = 4.12.”
The factor t for transmission of optical photons from the
phosphor to the silicon was evaluated for a fiber with n, = 1.81
and n, = 1.48 (numerical aperture = 1.04), core diameter 4.7 pm Table IV. Quantum efficiency values for different screen
and fiber pitch 5.7 pm (packing fraction = 4.7* / S.7* = 0.68), to thicknesses.
be t = 0.08.
The quantum efficiency of the CCD for detection of light at 545 nm is approximately cx = 0.22.”
4.3. DQEs
The effect of screenthickness on the DQE is shown in Fig. 8. A 10 pm protective overcoat is assumedhere. We have also
System
Phosphor-Fiber-Silicon
MTF’s
1 .o
1 .o
0.8
0.8
0.6
k
*
MTF
No warcoot
10 micron
20 mlsron
ovwsool
ov.rcoot
-
-
-
0.6
k
=
0.4
0.4
0.2
0.2
0.0
0.0
\
0
10
Spatial
20
Frequency
30
40
(1 /mm)
I
0
10
Spatial
.
‘I-
--
--,--
20
Frequency
30
1
40
(1 /mm)
Figure 6. System MTFs for a 40 pm thick screen with a 10
Figure 7. The effect of overcoat thickness on the phosphor-
ym thick overcoat.
fiber-silicon MTF for a 40 pm thick screen.
SPIE Vol. 1896 Physics of Medical
Imaging (I 993) I89
System
1 .o
0.8
w
go4=0.6.
I
6.
0.0
’ I
-
I.
1
20 microns40 rnitxmll 60 mlcrDna- - -
0
-3
\’
\\
\
’ ’ ’ ’ * ’
10
15
5
Spatial
I.
1.0
w
0
o
---- -s
--\\
0.2
’ I
System
DQE
0.8
-
0.6
-
0.4
-
0.2
-
1
0.0
20
Frequency
25
30
Spatial
(1 /mm)
*
I
’ I
’ I
MLloversocll
10 micron ovwsoot 20 micron 0vwsooL
-
I ,
10
15
5
0
35
I
DQE
20
Frequency
.
- -
. ,
30
35
25
(1 /mm)
Figure 8. The effect of screenthickness on the system DQE
Figure 9. The effect of overcoat thickness on system DQE
for a 10 pm thick overcoat.
for a 40 pm thick screen.
estimated the Swank factor of the screen for 20, 40 and 60 pm
screens to be 0.6, 0.65, and 0.7, respectively.6*7We see that
while a thicker screenimproves the quantum efficiency and thus
the DQE at low frequencies,a thinner screenactually improves
the DQE at high spatial frequencies. Fig. 9 shows the effect of
the thickness of the protective overcoat on the system DQE.
Once again we see that there is a substantial effect due to the
overcoat thickness.
Fig. 10 is a plot of the DQE vs. the system stage number
(determined with Eq. (25)) for a 40 pm screen with a 10 pm
overcoat. The drop in DQE from stage 1 to stage2 is due to the
Swank factor.
DQE vs.
1.0
[
I
Stage
I
I
0.8
#
I
I
-I
I-Omm
-, I - 10 mm-, I - 20 mm
- -
I
I
i
-.
0.2
0.0
-
---_
\
k
I
I
I
.
--
.
\ \
\
I
2
3
4
5
6
7
1
This display of the DQE, however, does not provide any
insight as to the origins of the DQE degradation. For instance,
Stage #
it does not indicate that the cause of the reduced DQE with Figure 10. DQE vs. stage # for a 40 pm thick screenand a
frequency is the optical scatter in the phosphor. This becomes 10 pm thick overcoat.
a greaterlimitation in more sophisticatedsystemswith additional
scattering processes. These limitations can be overcome by
exnmining the spatial-frequency dependentnomogram.
4.4. Spatial-Frequency
Dependent Nomograms
Figs. 11, 12, and 13 show nomograms for 20,40, and 60 pm screenswith 10 ym overcoats. The solid lines representthe
conventional DC nomograms, while the broken lines represent the spatial-frequency dependent nomograms. These figures do
not show the effect of the Swank factor as the nomogram does not include this effect. However, they do have the advantage
that they show much more clearly the effects of the various stageson the high-frequency information content. For example, they
show clearly the importance of the MTF of the phosphor-fiber-silicon combination in the splitting of the three lines between
stages 2 ‘and 3. At any frequency-independentstages, the lines ‘are parallel to one another due lo the log scale, while at
frequency-dependentstages,the lines diverge.
Fig. 13 shows a very significant loss of information at 20 mtn.’ for the 60 pm screen. It also shows that misleading
conclusions can result from the use of a DC nomogram. The zero-frequency analysis suggeststhat the system in Fig. 13 is
90 I S/'/E Vol. 1896 Physics of
h*/cdica//magin (1993)
quantum limited at the x-ray detection stage. However, Fig. 13 shows that this is true only up to spatial frequencies of
approximately 10 mm-‘.
Frequency-Dependent
I
Frequency-Dependent
Nomogram
I
I
1
Nomogram
!
I-Omm-,
, - 10
I-ZOrnm
-1
--
mm-(
,
:
W
5
=k
0.01
t
12
I
I
I
I
I
1
3
4
5
6
7
=a!=
0.01
1
I
12
I
I
I
I
1
3
4
5
6
7
Stage
#
Figure 11. Nomogram for a 20 pm thick screen with a 10
Stage
#
Figure 12. Nomogram for a 40 pm thick screen with a 10
pm thick overcoat.
ym thick overcoat.
Frequency-Dependent
Nomogram
o 10000
c
f-Omm-,
6
5
_-F
w
0
al
z
100
10
\
--\
1
=k
7
\
\
1:
\
---.
W
Ti
-
1000
.
0.1
0.01
12
.
.
,
I
I
I
3
4
5
6
.
.
7
Stage
#
Figure 13. Nomogram for a 60 pm thick screen with a 10
pm thick overcoat.
S.CONCLUSIONS
We have shown the utility of the cascadedsystem approach to system design. By brertking down the pCT detector system
into a series of simple mechanisms,an expression for the system spatial-frequency dependentDQE was obtained.
In addition, we have developed an expression for the spatial-frequency dependentnomogram, and used it to understand
signal and noise transfer characteristicsat each stage of the system. This approach is particularly useful for understandingthe
origins of DQE degradation,and can be used to m,akeappropriate design choices of system parameterssuch as optical coupling
efficiencies. It also eliminates the d&angerof the DC nomogram leading to misleading conclusions at high spatial frequencies.
The limitations of the nomogram approach include the fact that iI does not addressthe effect of other additive system noises,
or the effect of the Swank factor.
We have shown that there is a significant improvement in the MTF of the phosphor due to the finite acceptanceangle
defined by the fiber numerical aperture as well as the angle-dependenttransmittances al the screen-fiber and fiber-silicon
SPIE Vol. 1896 Physics of Medical
imaging
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I91
boundaries. This phosphor-fiber-silicon MTF is the most significant factor degrading the information content at high spatial
frequencies.We have also shown the dependenceof the DQE on screenthicknessand on the thicknessof the protective overcoat.
6. ACKNOWLEDGEMENTS
O”estionsof Dr. D.W. Holdsworth, Dr. A. Fenster and
The authors would like to acknowledge the helpful commentsand su,,
M. Drangova, and are grateful to the Medical ResearchCouncil of Canada for financial support. The first author is supported
by a Medical ResearchCouncil of Canada Studentship.
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